Recently the differential information-geometry analysis of quantum
fidelity in many-body systems displayed the intimate correlation
between quantum phase transition and the singularity of fidelity
between the states across the transition point [1]}. In
these papers the quantum geometric tensor, which is intimately
related to the degeneracy of the ground state, was introduced for
the determination of the fidelity. As shown in these papers, the
imaginary part of this tensor actually described the curvature
two-form whose holonomy is the Berry phase, and the degeneracy of
the ground state would induce the singularity of fidelity
[1]}. However, in our opinion, it seems not transparent to
directly define the geometric phase in this coupling-parameter space
since a cyclic evolution may be difficult to construct. Moreover,
the explicit expression for the ground state is necessary for the
construction of this tensor, which in general is difficult.
Furthermore it is also unknown in the case where the degeneracy of
the ground state is broken. In Ref. [3]}, the Bargmann
phase, a generalization of the Berry phase, has also been
constructed for detecting the phase transition in many-body systems.
However the connection between degeneracy of the ground state and
the Bargmann phase was unclear in this case since there was a lack
of simple interpretational underpinnings for the Bargmann phase in
terms of physical adiabatic processes [3]}.
